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# System prepended metadata

title: Research Associate Take-Home Exercise

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#  Research Associate Take-Home Exercise

The goal of this exercise is to get a sense for the type of work we do on the team. It should only take a few hours (please let us know if it's taking longer than that). Please explain your reasoning as you go. If you have any questions, please reach out to dan@paradigm.xyz or frankie@paradigm.xyz. 

1. The [VRGDA paper](https://www.paradigm.xyz/2022/08/vrgda) describes a mechanism for selling NFTs over time, while targeting a particular issuance schedule. We'd like to extend this mechanism to sell fungible tokens. 

    Suppose our auction targets sells token ABC for USD. We are targeting a sales rate of $r$ per day (meaning if ABC tokens are sold at exactly that rate, the price does not change). We also want the marginal price of ABC to decay exponentially if no tokens are sold. Let $k$ be the rate of decay per day (that is, if price falls $10\%$ per day, $k=0.1$), $p_0$ be the initial price of the auction, $t$ be the amount of time in days since the beginning of the auction, and $n$ be the amount of ABC sold to date. 
 
    a. What is the formula for the marginal price of ABC in USD, as a function of the above variables?
    
    b. What is the formula for the amount of USD you have to spend to buy some amount $q$ of ABC?
    
    c. What is the formula for the amount of ABC you would receive if you paid $p$ USD? 
    
2. Please implement a tool that demonstrates the working of this mechanism in your platform of choice (i.e. Python, R, Desmos, Google Sheets, etc). 

3. The continuous VRGDA mechanism is similar to a [TWAMM](https://www.paradigm.xyz/2021/07/twamm). Please explain the similarities and differences, and describe when you'd use one mechanism or the other. 

4. [Art Gobblers](https://www.paradigm.xyz/2022/09/artgobblers) is an example of a mechanism that uses VRGDAs in production. Please generate some analysis of how the mechanism has behaved, based on onchain data. 